Kinetic Growth Aggregation (Leath Algorithm) Approach for Continuum Percolation Model

2009 ◽  
Author(s):  
Keizo Yamamoto ◽  
Hiroyuki Yoshinaga ◽  
Sasuke Miyazima ◽  
Theodore E. Simos ◽  
George Psihoyios ◽  
...  
Keyword(s):  
Percolation Model ◽  
Continuum Percolation ◽  
Kinetic Growth

Co-Existence of the occupied and vacant phase in boolean models in three or more dimensions

1997 ◽  
Vol 29 (4) ◽  
pp. 878-889 ◽  
Author(s):  
Anish Sarkar
Keyword(s):  
Poisson Process ◽  
Boolean Model ◽  
Percolation Model ◽  
Continuum Percolation ◽  
Boolean Models ◽  
Whole Space ◽  
Critical Intensity ◽  
The Way

Consider a continuum percolation model in which, at each point of ad-dimensional Poisson process of rate λ, a ball of radius 1 is centred. We show that, for anyd≧ 3, there exists a phase where both the regions, occupied and vacant, contain unbounded components. The proof uses the concept of enhancement for the Boolean model, and along the way we prove that the critical intensity of a Boolean model defined on a slab is strictly larger than the critical intensity of a Boolean model defined on the whole space.

scholarly journals Line-of-Sight Percolation

2009 ◽  
Vol 18 (1-2) ◽  
pp. 83-106 ◽  
Author(s):  
BÉLA BOLLOBÁS ◽  
SVANTE JANSON ◽  
OLIVER RIORDAN
Keyword(s):  
Random Walk ◽  
Higher Dimensions ◽  
Percolation Model ◽  
The Other ◽  
Branching Random Walk ◽  
Line Of Sight ◽  
Continuum Percolation ◽  
Critical Probability ◽  
Site Percolation ◽  
Vertex Set

Given ω ≥ 1, let $\Z^2_{(\omega)}$ be the graph with vertex set $\Z^2$ in which two vertices are joined if they agree in one coordinate and differ by at most ω in the other. (Thus $\Z^2_{(1)}$ is precisely $\Z^2$.) Let pc(ω) be the critical probability for site percolation on $\Z^2_{(\omega)}$. Extending recent results of Frieze, Kleinberg, Ravi and Debany, we show that limω→∞ωpc(ω)=log(3/2). We also prove analogues of this result for the n-by-n grid and in higher dimensions, the latter involving interesting connections to Gilbert's continuum percolation model. To prove our results, we explore the component of the origin in a certain non-standard way, and show that this exploration is well approximated by a certain branching random walk.

Scaling and Continuum Percolation Model for Enzyme-Catalyzed Gel Degradation

2007 ◽  
Vol 98 (22) ◽  
Author(s):  
D. Lairez ◽  
J.-P. Carton ◽  
G. Zalczer ◽  
J. Pelta
Keyword(s):  
Percolation Model ◽  
Continuum Percolation

scholarly journals Fractal structure of equipotential curves on a continuum percolation model

2012 ◽  
Vol 391 (23) ◽  
pp. 5802-5809 ◽  
Author(s):  
Shigeki Matsutani ◽  
Yoshiyuki Shimosako ◽  
Yunhong Wang
Keyword(s):  
Fractal Structure ◽  
Percolation Model ◽  
Continuum Percolation

scholarly journals Percolation on the Information-Theoretically Secure Signal to Interference Ratio Graph

2014 ◽  
Vol 51 (04) ◽  
pp. 910-920
Author(s):  
Rahul Vaze ◽  
Srikanth Iyer
Keyword(s):  
Point Processes ◽  
Interference Suppression ◽  
Percolation Model ◽  
The Other ◽  
Continuum Percolation ◽  
Directed Edge ◽  
Connected Component ◽  
Poisson Point Processes ◽  
Subcritical Regime ◽  
Critical Intensity

We consider a continuum percolation model consisting of two types of nodes, namely legitimate and eavesdropper nodes, distributed according to independent Poisson point processes in R 2 of intensities λ and λ E , respectively. A directed edge from one legitimate node A to another legitimate node B exists provided that the strength of the signal transmitted from node A that is received at node B is higher than that received at any eavesdropper node. The strength of the signal received at a node from a legitimate node depends not only on the distance between these nodes, but also on the location of the other legitimate nodes and an interference suppression parameter γ. The graph is said to percolate when there exists an infinitely connected component. We show that for any finite intensity λ E of eavesdropper nodes, there exists a critical intensity λ c < ∞ such that for all λ > λ c the graph percolates for sufficiently small values of the interference parameter. Furthermore, for the subcritical regime, we show that there exists a λ0 such that for all λ < λ0 ≤ λ c a suitable graph defined over eavesdropper node connections percolates that precludes percolation in the graphs formed by the legitimate nodes.

Comment on “Exact solution of a one-dimensional continuum percolation model”

1997 ◽  
Vol 56 (5) ◽  
pp. 6206-6207 ◽  
Author(s):  
Luis A. Pugnaloni ◽  
Ricardo D. Gianotti ◽  
Fernando Vericat
Keyword(s):  
Exact Solution ◽  
Percolation Model ◽  
Continuum Percolation ◽  
One Dimensional

Behavior of the supercritical phase of a continuum percolation model on ℝd

1993 ◽  
Vol 30 (2) ◽  
pp. 382-396 ◽  
Author(s):  
Hideki Tanemura
Keyword(s):  
Positive Constant ◽  
Effective Conductivity ◽  
Critical Value ◽  
Percolation Model ◽  
Continuum Percolation ◽  
Supercritical Case ◽  
Supercritical Phase ◽  
Renormalization Technique

A continuum percolation model onis considered. Using a renormalization technique developed by Grimmett and Marstrand, we show a continuum analogue of their results. We prove the critical value of the percolation equals the limit of the critical value of a slice as the thickness of the slice tends to infinity. We also prove that the effective conductivity in the model is bounded from below by a positive constant in the supercritical case.

scholarly journals Homological Percolation: The Formation of Giant k-Cycles

2020 ◽  
Author(s):  
Omer Bobrowski ◽  
Primoz Skraba
Keyword(s):  
Thermodynamic Limit ◽  
Exponential Decay ◽  
Algebraic Topology ◽  
Percolation Model ◽  
Critical Values ◽  
Giant Component ◽  
Connected Components ◽  
Continuum Percolation ◽  
Dimensional Torus ◽  
Higher Dimensional

Abstract In this paper we introduce and study a higher dimensional analogue of the giant component in continuum percolation. Using the language of algebraic topology, we define the notion of giant $k$-dimensional cycles (with $0$-cycles being connected components). Considering a continuum percolation model in the flat $d$-dimensional torus, we show that all the giant $k$-cycles ($1\le k \le d-1$) appear in the regime known as the thermodynamic limit. We also prove that the thresholds for the emergence of the giant $k$-cycles are increasing in $k$ and are tightly related to the critical values in continuum percolation. Finally, we provide bounds for the exponential decay of the probabilities of giant cycles appearing.

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